# \(\Lambda_{24}\) Leech lattice-shell code[1]

## Description

Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code, and recursively taking their kissing configurations yields the \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes [2]; all codes are optimal and unique for their parameters [3,4].

## Protection

Smallest-shell code yields an optimal solution to the kissing problem in 24D [3]. This code saturates the Levenshtein bound [5–7][8; pg. 337] and is unique up to equivalence [3].

## Parent

## Cousins

- \(\Lambda_{24}\) Leech lattice code
- Spherical sharp configuration — The smallest-shell \((24,196560,1)\) code is a spherical sharp configuration [9,10]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are also sharp configurations [3,4,10,11][12; Table 1].
- Spherical design — Smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [8; Ch. 3]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are spherical 7- and 5-designs, respectively [3,4,10,11][12; Table 1].

## References

- [1]
- J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
- [2]
- P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
- [3]
- E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
- [4]
- H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
- [5]
- V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
- [6]
- V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.
- [7]
- A. M. Odlyzko and N. J. A. Sloane, “New bounds on the number of unit spheres that can touch a unit sphere in n dimensions”, Journal of Combinatorial Theory, Series A 26, 210 (1979) DOI
- [8]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [9]
- Andreev, N. N. Location of points on a sphere with minimal energy. (Russian) Tr. Mat. Inst. Steklova 219 (1997), Teor. Priblizh. Garmon. Anal., 27–31; translation in Proc. Steklov Inst. Math. 1997, no. 4(219), 20–24
- [10]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [11]
- R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
- [12]
- H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202

## Page edit log

- Victor V. Albert (2022-11-17) — most recent

## Cite as:

“\(\Lambda_{24}\) Leech lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech_shell